Lecture Notes on Law and Economics: Theory
Elliott Ash
July 29, 2015
These lecture notes are an attempt to synthesize the various source materials for this course
under a single framework with unified notation.
1
Introduction to Legal Incentives
This section presents basic models of the key problems solved by property law, contract law, tort
law, and criminal law.
1.1
Property Law
Consider a landowner A and his neighbor, B. A chooses a binary action, x ∈ {0, 1}. The classic
example is that A is a rancher and B is a farmer. x = 1 means he builds a fence, which would
protect B’s crops from the cattle trodding over them. x = 0 means the fence is not built. A and
B begin with endowments ωA and ωB . A monetary transfer between the parties is represented by
w. The cost of x to A is parametrized by a and the benefit of x to B is parametrized by b. The
payoff functions for A and B are
uA = ωA + w − ax
and
uB = ωB − w − b(1 − x).
In the classic story where the rancher has to build a fence, we think of a as the cost of building
the fence, while b is the damage done to the farmer’s crops if the fence is not built.
Now consider two different legal regimes. In Regime 1 (“Rancher’s Rights”), A is not responsible
for any damages caused by his cattle. In Regime 2 (“Farmer’s Rights”), A has to pay for any
damages caused by his cattle.
In Regime 1, the payoffs for A and B for x ∈ {0, 1} are:
1
uA (x = 0, w = 0) = ωA
uA (x = 1, w = 0) = ωA − a
uB (x = 0, w = 0) = ωB − b
uB (x = 1, w = 0) = ωB
In the absence of an agreement, A chooses x = 0 because ωA ≥ ωA − a.
In Regime 2, the payoffs are
uA (x = 0, w = 0) = ωA − b
uA (x = 1, w = 0) = ωA − a
uB (x = 0, w = 0) = ωB
uB (x = 1, w = 0) = ωB
reflecting that A is going to pay B’s damages when x = 0. In this case, A chooses x = 1 if
a < b and x = 0 if a > b. Note that Regime 2 results in the socially optimal outcome given the
structure of the problem, since A internalizes the costs of the externality.
Now return to Regime 1, but allow bargaining between A and B. The neighbors will write a
contract specifying a fence choice x̃ and a transfer w. In particular, B can pay A to build the
fence. In order to persuade A to build the fence, B must offer w ≥ a. But B stands to gain only
if w ≤ b. The range of w for which both A and B benefit from the fence is w ∈ [a, b], but this
interval exists only when a ≤ b. Therefore an agreement to build the fence will only occur when
the benefit of the fence b is greater than or equal to the cost of the fence a. If a < b, not building
the fence is not an equilibrium because B would improve his payoff by offering w ≥ a and having
the fence built. If a > b, building the fence is not an equilibrium because B would be paying
w > b, in which case he would get a better payoff by abandoning the deal and allowing his crops
2
to be trampled. If a = b, A and B are indifferent between having or not having a fence (and
therefore an agreement). This coincides with choosing x to optimize the social welfare function
W = uA + uB = ωA + ωB + (b − a − 1)x.
Now return to Regime 2 allowing for an agreement (x̃, w). In this case A must pay B if the
crops are trampled. As with the case where no bargaining was allowed, A will build the fence
when b > a to avoid paying damages. When a < b, he won’t build a fence and will go ahead and
pay the damages. Therefore, the outcome is the same as under Regime 1. This again results in
the social optimal outcome on x.
In the absence of transaction costs, default property rights don’t matter. This result is known
as the Coase Theorem.
1.2
Contract Law
Consider a seller A and a buyer B. A promises to sell to B a good with quality x ≥ 0 at price
w, but B will observe the quality only after paying w. The cost of quality for A is ax, while the
benefit of quality to B is b(x), where we assume b(0) = 0, b0 (·) > 0, and b00 (·) < 0. The payoff
from the transaction for the seller is
uA = w − ax
and the buyer’s utility is given by
uB = b(x) − w
We assume the outside options for each agent are such that payoffs are zero without a contract.
So the transaction is mutually beneficial for (x, w) such that w ≥ ax and b(x) ≥ w, which can
be written as ax ≤ w ≤ b(x).
The parties have written a contract, and B has paid w. What will A do? In the absence
of any legal incentives, A chooses x = 0 and earns payoff w; B receives the good and observes
quality x = 0 and is left with payoff −w. Looking ahead from the beginning of the relationship,
B will see this coming and refuse to enter the transaction. Both parties get a zero payoff.
Now take the case of enforceable contracts. After receiving the good, B observes quality x
and see if it corresponds to the contracted level x̃. If x < x̃, B can file a claim for breach of
contract. With the standard rule of expectations damages, A will be forced to compensate B for
any difference between his contracted payoff level b(x̃) − w and the actual payoff b(x) − w. That
is, A will pay damages z such that
b(x̃) − w + z = b(x) − w
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which means
z(x) = b(x̃) − b(x).
Under this system of legal incentives, the seller’s payoff as a function of x when x ≤ x̃ is
uA = w − ax − z
= w − ax + b(x) − b(x̃).
A0 s choice of x maximizes this function, which gives
b0 (x) = a
Which is the same quality level as if he just followed the contract. Define x∗ as the quality level
satisfying this condition. This level is socially efficient. At the social optimum, the marginal
benefit of quality equals the marginal cost, which is satisfied under contract enforcement with
expectations damages.
The equilbrium contracts are the set of contracts (x̃, w̃) where x̃ = x∗ and w satisfies the
participation constraints
w − ax∗ ≥ 0
and
b(x∗ ) − w ≥ 0
which means
ax∗ ≤ w ≤ b(x∗ ).
This means that a mutually beneficial contract exists, and the transaction will occur, when
ax∗ ≤ b(x∗ )
This is better than the situation without legal enforcement of contracts, where the transaction
can never occur in equilibrium.
Example. Let b(x) = β1 xβ with β ∈ (0, 1). What is the contracted quality level x̃ without
contract enforcement? With contract enforcement? What are the damages paid for quality
x < x̃? What is the range of w for which a Pareto optimal transaction exists?
Solution. Without contract enforcement, the buyer knows that the seller will renege on any
contracted level x̃ > 0 and choose x = 0. So no contract occurs. With contract enforcement,
the buyer chooses x̃ = x∗ , the socially optimal level where b0 (x) = a. Here, b0 (x) = xβ−1 . So we
4
have
xβ−1 = a
1
x∗ = a β−1 .
How do we know he chooses this level? It is the only equilibrium with free negotiation. If the buyer
proposed x̃ < x∗ , he could increase the size of the surplus by choosing x̃0 such that x̃ < x̃0 ≤ x∗ ,
which for constant w would increase the payoff to the buyer without decreasing the payoff to the
seller. The same is true for x̃ > x∗ .
Under expectations damages, the seller will pay z(x) = b(x̃) − b(x) for choosing x < x̃. For
the functional for we have
z(x) = b(x̃) − b(x)
1 β 1 β
x̃ − x
=
β
β
1
for x̃ = x∗ = a β−1 , this is
z(x) =
1
1
1 β−1
(a )β − xβ
β
β
β
a β−1 − xβ
=
β
The range of w for which a mutually beneficial transaction exists is
w ∈ [ax∗ , b(x∗ )]
1
1
1
∈ [a1 a β−1 , (a β−1 )β ]
β
β
w
∗
∈ [a
β
β−1
a β−1
,
]
β
Because β < 1, the second term in the interval is greater than the first term. A more concave
utility functionb(x) means a larger contracting surplus.
1.3
Tort Law
Consider an injurer A and a victim B. A is performing some activity, such as driving, that poses
some risk of harm b to B. A chooses some level of precaution x ≥ 0 that has a cost a but reduces
5
the probability that the harm to B occurs. The payoff function for A is
uA = ωA − ax
where ωA is an endowment. The harm occurs with probability 1 − p(x), where p(x) is strictly
increasing and strictly concave in x. If the harm does not occur B receives
uB = ωB
and if the harm does occur he receives
uB = ωB − b
where ωB is again an endowment. The expected payoff to B is
E(uB ) = ωB − (1 − p(x))b
Without tort law, A gives no concern to the externality imposed on B – he chooses x = 0
and earns his endowment ωA . B’s expected utility is ωB − (1 − p(0))b. Under the traditional
compensation rule in tort, A just has to compensate B for any harm inccurred. Under tort law,
B’s utility does not depend on whether the harm occurs because he is perfectly compensated
when it does occur:
uB = ωB
The injurer now bears the risk. Without the harm, he receives
uA = ωA − ax
and with the harm he receives
uA = ωA − ax − b
The expected utility is
E(uA ) = p(x)(ωA − ax) + (1 − p(x))(ωA − ax − b)
= ωA − ax − (1 − p(x))b
Choosing x to maximize expected utility gives
a = p0 (x)b
6
let x∗ be the precaution level that satisfies this condition. This is the socially optimal precaution
level.
Note that under tort law incentives, A will not engage in the activity unless there is a net
social benefit to the activity.
Example. Let p(x) = 1 − xρ . What is the precaution level and resulting utilities without tort
law? With tort law?
Solution. Without tort law, x = 0. uA = ωA and
E(uB ) = ωB − (1 − p(x))b
ρ
= ωB − (1 − (1 − ))b
x
ρ
= ωB − b
x
ρ
E(uB |x = 0) = ωB − b
0
→ −∞
Fortunately for B, tort law relieves him of risk: uB = ωB . The injurer now has
E(uA |x) = ωA − ax − (1 − p(x))b
ρ
= ωA − ax − (1 − (1 − ))b
x
ρb
= ωA − ax −
x
Take the derivative for x:
0 = −a +
ax2 = ρb
r
x∗ =
ρb
x2
ρb
a
Which gives expected utility
r
E(uA |x =
r
ρb
ρb
ρb
) = ωA − a
−q
a
a
ρb
a
p
= ωA − 2 ρab
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1.4
Criminal Law
Consider a criminal A who chooses a level of criminal activity with severity x ≥ 0. The activity
accrues private benefits ax to the criminal but imposes a negative externality on the rest of the
society – the classic example is tax evasion, where x is how much tax to evade in dollars. The
criminal is caught with some probability p(x, y), where y is the amount of tax money invested by
the government in police and other law enforcement efforts. The assumptions are px > 0, py > 0,
pxx < 0,pyy < 0, pxy > 0 (where subscripts denote partial derivatives). The criminal law specifies
a punishment b(x) that the criminal bears in the event of detection, were bx > 0.
When he is not detected, the criminal’s utility function is
uA = ax.
When he is detected and punished, his utility is
uA = ax − b(x).
The expected utility is
E(uA ) = (1 − p(x, y))ax + p(x, y)(ax − b(x)
= ax − p(x, y)b(x)
The criminal maximizes this equation – taking y as given, the optimality condition is
a = px (·)b(·) + p(·)bx (·)
A higher marginal benefit to crime a will increase the chosen crime level, while an increase in the
punishment function b(·) will results in less crime.
Let x∗ (y) be the criminal’s optimizing crime choice given the enforcement expenditure y.
Let’s say that society’s welfare function does not depend on whether the criminal is caught for
his crimes – compensation is impossible:
W = ω − y − x∗ (y)
In that case crime enforcement expenditure have no social benefit beyong reducing the crime level
x. In particular, the social optimum requires
1 = −x∗y (y)
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That is, the marginal cost of tax expenditures on enforcement (assumed to equal one in this case)
equals the marginal reduction in crime.
Example. Let b(x) = x and Let p(x, y) = x1/2 y. Solve for the crime level as a function of
y, and then solve for the optimal enforcement expenditure y.
Solution. We have bx = 1 and
px =
∂p(·)
1 1
= x− 2 y
∂x
2
Using the criminal’s optimality condition
a = px (·)b(·) + p(·)bx (·)
gives
1
1 1
a = [ x− 2 y]x + x 2 y
2
3y 1
a =
x2
2
1
2a
= x2
3y
4a2
x∗ (y) =
9y 2
Which is the criminal’s best response function to the government enforcement expenditure, decreasing in y, as desired. To get the the optimal government expenditure, take the social welfare
function
W (y) = ω − y − x∗ (y)
4a2
= ω−y− 2
9y
and take the derivative for y:
∂W
∂y
= −1 +
8a2
9y 3
8 2
=
a
9
8 1 2
= ( )3 a3
9
1 =
y3
y∗
8a2
9y 3
9
gives the government’s optimal crime enforcement expenditure.
Now plug this back into the criminal’s best response function:
4a2 −2
y
9
4 2 8 1 2 −2
=
a (( ) 3 a 3 )
9
9
4 2 8 −2 −4
a · ( ) 3a 3
=
9
9
− 23 23
= 3 a
x∗ (y) =
x∗
The probability of detection x1/2 y is
1
p(x, y) = x 2 y
2 1
2
8 1 2
= (3− 3 a 3 ) 2 (( ) 3 a 3 )
9
− 13 13 8 13 23
= 3 a ( ) a
9
2
∗ ∗
a
p(x , y ) =
3
Expected utility for the criminal is
E(uA ) = ax − p(x, y)b(x)
2
2
2
2
2
= a(3− 3 a 3 ) − a(3− 3 a 3 )
3
√
3
2
5
2 3 5
= 3− 3 a 3 −
a3
9
√
3
3 5
a3
=
9
And finally, social welfare for society is given by
W ∗ = ω − y ∗ − x∗
2
2
8 1 2
= ω − ( ) 3 a 3 − 3− 3 a 3
9
1
2
= ω − 33 a3
Note how these items change with the benefit from crime, a.
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2
Rules Versus Taxes
Most textbook discussions of law and economics (including Cooter and Ulen) ignore an important
problem with the use of legal rules. In particular: Why use legal rules rather than taxes? The
standard models in public economics have figured out how to deal with externalities by implementing an optimal Pigouvian tax. This section discusses reasons to use rules rather than taxes,
an analysis motivated by O’Flaherty (2005, ch. 8 sec. 2).
2.1
Pigouvian Taxes versus Legal Rules
Now some excerpts from O’Flaherty, 2005, Chapter 8:
• Neoclassical economists generally like prices a lot more than rules (by rules here I mean
rules about quantities—how much you can spit on the street—not rules about prices). The
reasoning should be familiar: if the government or the firm has enough information to set
a price so it reflects the true (marginal) cost of an activity, then the only instances of an
activity that will occur are those in which benefits exceed costs (to everyone), and these are
precisely the instances that should occur if no further potential Pareto im- provements are to
be possible. Rules, in contrast, appear to neoclassical economists to be blunt instruments:
they don’t differentiate well between those instances of an activity that should be done and
those that should not.
• The neoclassical case against rules, however, is almost surely overstated. It compares ideal
prices—the prices you could set only with excellent information—with actual rules—rules
set in a context of poor information. In fact, if a government (or a firm) has the superb
information it needs to set precisely the right prices, then it can set rules that work just as
well as prices: it can permit those instances in which benefits exceed costs, and prohibit the
others. When information is very good, rules work just as well as prices. The real question,
then, is whether rules work better than prices when the information the government has is
poor. The answer here is the usual, “It depends.”
• Sometimes collecting money is simply infeasible under current technology, as at a traffic
intersection or when a building is on fire. Other times the external harm of an act is so
great that the optimal number of times for the act to be committed is clearly zero. Arson
and murder (except in self-defense) are examples of this case.
• Rules work better than prices when the marginal external cost of what the public is doing
is very sensitive to the quantity of what the public is doing, and when the marginal private
11
benefit is not very sensitive to the quantity. Prices work better than rules in the opposite
cases.
• For instance, a traffic intersection should be governed by rules because the marginal external
cost of one car going through—about zero—is a lot less than the marginal external cost of
two cars going through at the same time—a collision—while the marginal private benefit
of each passage does not vary as much.
• Consider a small cruise ship with enough lifeboats for 100 people. The ship has 20 crew
members... Thus each passenger beyond the eightieth endangers an additional crew member, while the first 80 passengers pose no such danger. The marginal social cost of carrying
passengers beyond a total of 80 is much greater than the marginal social cost of carrying 80
passengers. If you’re running the cruise ship ... you’ll set a rule: no more than 80 passengers. You know the quantity of people beyond which marginal passengers harm the crew,
but you don’t know for sure how many passengers you’ll have at a given price. If you set a
price without a quantity rule and more than 80 passengers book passage, the consequences
could be tragic. So when you don’t know demand but you do know a quantity at which
marginal social cost changes (80), it makes sense to use a rule rather than a price.
• In contrast, consider a copy machine in a public library. The library must replace the
machine’s ink, toner, and paper when they run out. Since all of these items are for sale in
regular stores, the cost of a copy to the library—the marginal social cost—is pretty much
the same no matter how many copies have been made in a given day. There’s no magic
number of copies after which the marginal cost suddenly jumps up. So setting a rule such
as “Only 80 copies a day” would be a bad idea for the library. If patrons happened to want
to make 200 copies some day and were willing to pay for the ink, toner, and paper for each
of them, the rule would preclude a potential Pareto improvement. Setting the price equal
to marginal social cost makes a lot more sense. Then no matter how many copies patrons
wanted to make at that price, the outcome would be Pareto optimal.
• Thus when you know what the marginal social cost is going to be, as at the library, set a
price. When you don’t know what marginal social cost is go- ing to be but do know where
it changes, as on the cruise ship, use a rule. For situations in between, use a rule for those
more like the cruise ship case and a price or tax for those more like the library case.
• The second argument for using rules rather than Pigouvian taxes concentrates on enforcement costs—how hard it is to find out what people have done in order to tax or punish them
appropriately. Community reporting is cheap and often important, and it’s much easier for
12
neighbors and the community in general to know whether someone is breaking a rule than
whether she has paid appropriate taxes.
• For instance, consider Sunday liquor sales. The externalities these cause could be mitigated
by either putting an extra tax on them or prohibiting them. If sales are prohibited, then
it’s easy for neighbors, clergy, and passersby to tell whether a liquor store is in compliance,
and easy for a judge or jury to decide guilt or innocence. In contrast, a passerby on her
way to church on a Sunday morning has no way to tell whether anyone is actually paying
the appropriate Pigouvian surcharge on liquor.... Thus, so far as enforcement is concerned,
rules are likely to be a cheaper and more effective way of dealing with Sunday liquor sales
than are Pigouvian taxes.
• In general, then, when community monitoring is cheap and easy, and when tax collectors
are likely to be lazy or corrupt, rules will often work better than taxes to enforce the desired
behavior. Some other examples are hunting and fishing limits, rules on where buildings can
be built, height limitations, restaurant and bar closing hours, noise ordinances, and laws on
endangering the welfare of children.
2.2
Using taxes in the basic models of legal incentives
The market failures detailed in the basic models of Section 2 can all be eliminated through
taxes rather than by legal rules. Basically, the Pigouvian approach of taxing externalities at their
marginal external cost works provided the government has perfect enforcement power and perfect
information about the payoffs. In the property model, the government can give efficient fencing
incentives to the rancher simply by imposing a tax b for not building a fence (or a subsidy of b for
building a fence). In the contract model, similarly, the government gives efficient incentives for
product quality by setting a marginal subsidy of b0 (x) to the seller for quality (this is equivalent
to a marginal tax of b0 (x) on lower quality). In tort law, a marginal tax of p0 (x)b as precaution
decreases (a marginal subsidy of p0 (x)b as precaution increases) gives efficient incentives. In
criminal law, finally, a 100% marginal tax on x is optimal.
For the four situations described in the basic legal incentives models, it’s clear why taxes usually
won’t work. Even if their were funds available to administer these taxes, it would be difficult if
not impossible to learn the payoffs. The rancher would disagree with the farmer’s assessment of b
in order to pay a lower tax. In the contract example, the government probably couldn’t measure
quality very effectively and wouldn’t have good information about b0 (x). In tort, similarly, the
government doesn’t know p(x) or b, especially when the harm doesn’t occur. In criminal law,
finally, the whole point is that the criminal cannot be taxed (punished) if not detected.
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2.3
Police Powers
These are relevant excerpts from O’Flaherty, 2008, Chapter 8, Section II:
• Because regulations are often a good idea, it’s not bad to have entities that can promulgate
them. Because circumstances vary and change rapidly, and regulations need to adapt, the
promulgating entities should have some discretion. Having someone around who has police
powers can make all of us who are subject to those powers better off.
• Discretion and the power to coerce, however, are a dangerous combination. An entity with
discretion and power can coerce all the rent for an entire city into its own hands simply
by extortion . . . If there are some investments that people will make only if they can
be rewarded for them later, then those investments will never be made. The coercive
entity could never convince people that anything would ever restrain its capacity enough
to let them harvest those rewards. Instead of making us all better off, police powers, if
untrammeled, could make us all worse off. . . . Police powers, therefore, are circumscribed
in many different ways.
• Procedural due process requires that governments follow some sort of already prescribed
tests before they exercise police powers against someone.
• Substantive due process requires that the rules be reasonably related to the purpose of police
powers—to promote the health, safety, morals, and general welfare of the community.
• Equal protection says that rules must be written and applied impersonally.
• The First Amendment protections of religion, speech, assembly, and po- litical activity also
circumscribe some police powers.
• The final constitutional check on police powers is known as the takings clause. . . . This
refers to an ancient practice called eminent domain. When a government is building a road
or a firehouse, it can force landowners to sell it the property it needs—but it must pay them
a fair price.
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3
Bargaining
This section presents the basics of bargaining theory that will come in handy for this course.
3.1
Nash Bargaining Solution
Bargaining problems represent situations in which:
• There is a conflict of interest about agreements.
• Individuals have the possibility of concluding a mutually beneficial agreement.
• No agreement may be imposed on any individual without his approval.
There are two players – parties to a negotiation. Suppose the players i = A, B have preferences
represented by a utility function ui (xi ) where xi is the outcome of the bargain for player i. We
can think of the outcomes as payments xi out of a fixed pie of size π, so xA + xB ≤ π, If no
agreement is reached, the players receive dA and dB , respectively. This is called the disagreement
outcome or “threat point.”
A pair of payoffs (u∗A , u∗B ) is a Nash bargaining solution if it solves the following problem:
max (uA − dA )(uB − dB )
uA ,uB
subject to the constraints:
xA + xB ≤ π
uA ≥ dA
uB ≥ dB
That is, it maximizes the product of the utility levels net of the disagreement utilities.
Example 1. Suppose there is $100 on the table to be split between two players (π = 100).
Players receive linear utility: u(xi ) = xi . If no agreement is reached, players do not receive
anything: dA = dB = 0. The feasibility constaint is x1 + x2 ≤ 100, while the participation
constraints are xA ≥ 0, xB ≥ 0. So the Nash bargaining problem is
max uA uB
uA ,uB
15
And we substitute ui = xi into the constraints:
uA + uB ≤ 100
uA ≥ 0
uB ≥ 0
The constraint binds at the optimum, so you can substitute uB = 100−uA . Maximizing uA (100−
uA ) gives u∗A = u∗B = 50.
Example 2. Same as Example 1, except dA = 20. The Nash bargaining problem is
max (uA − 20)uB
uA ,uB
subject to
uA + uB ≤ 100
uA ≥ 20
uB ≥ 0
As before, you can substitute uB = 100 − uA . Maximizing (uA − 20)(100 − uA ) gives u∗A =
60, u∗B = 40.
√
Example 3. Same as Example 1, except uB = xB . We account for the change in utility
function is to change the substitution for xi in the constraints. In particular, instead of xB = uB
we now have xB = u2B . The Nash bargaining problem is
max uA uB
uA ,uB
subject to
uA + u2B ≤ 100
uA ≥ 0
u2B ≥ 0
16
The constraint binds, so substitute uA = 100 − u2B and differentiate
(100 − u2B )uB
100uB − u3B
100 − 3u2B = 0
100
u2B =
r3
100
u∗B =
3
To get, uA , plug back into the feasibility constraint to get
r
uA = 100 − (
100 2 200
) =
3
3
The risk averse player gets a lower utility.
The Nash bargaining solution has two noteworthy features:
• Symmetry. If all agents are identical, then the gains are split equally.
• Pareto efficiency. There exists no feasible outcome in which one player is better off and the
other player is at least as well off.
Here are two alternative solution methods:
• Egalitarian solution. The gains from bargaining are split equally among the agents regardless
of the threat point.
• Utilitarian solution. The total utility is maximized:
max uA + uB
uA ,uB
These methods don’t account for the threat point. That partly explains why they’re less realistic
in explaining bargaining behavior.
3.2
The alternative-offers model of bargaining
Consider two agents A and B who have to decide how to split some surplus π.1 In law and
economics, this could be the surplus from a contracted investment, for example, or the terms of a
1
This discussion of alternating-offers bargaining is derived from Slantchev 2008,
http://slantchev.ucsd.edu/courses/gt/05-extensive-form.pdf starting at pg. 38.
17
available at
trial settlement. The rules of the game are that the agents take turns proposing a division of the
pie. Begin with player A as the proposer; he offers x ∈ [0, π] as the share he keeps. If B accepts
the offer, then an agreement is made; A receives x and B receives π − x. If B rejects, the roles
flip. B becomes the proposer and makes a counter-offer y ∈ [0, π]; if A accepts, B receives y
and A receives π − y. And so on.
Players discount the future with a common discount factor δ ∈ [0, 1]. While players disagree,
neither receives anything (which means that if they perpetually disagree then each player’s payoff
is zero). If some player agrees on a partition(x, π − x) at some time t, player A’s payoff is δ t x and
player B’s payoff is δ t (π − x). This formalizes the idea that it is better to obtain an agreement
sooner rather than later, due to impatience or due to the costs of bargaining/negotiations.
The subgame perfect Nash equilibrium is a set of equilibrium offers x∗ and y ∗ that satisfies
two conditions: (1) an equilibrium offer is always accepted, and (2) each player always makes the
same offer. Consider some arbitrary period t when A is making an offer x∗ to B. If B rejects,
he must offer y ∗ due to (2), and A will accept it, due to (1). Therefore the payoff to B for
rejecting an offer in equilibrium is δy ∗ (it is discounted by δ because he has to wait until the next
period). Therefore B will reject any offer π − x < δy ∗ and accept any offer π − x > δy ∗ . Further,
assume that players accept when indifferent, so B will accept when π − x = δy ∗ . So Condition
(1) requires that π − x∗ ≥ δy ∗ . But π − x∗ > δy ∗ would not be a best response for A because
he could offer π − x = δy ∗ and earn a higher payoff with B still accepting. In equilibrium, B is
indifferent between accepting and rejecting:
π − x∗ = δy ∗
and symmetrically, A must be indifferent between accepting and rejecting:
π − y ∗ = δx∗
You can use substitution with these equations to solve for x∗ and y ∗ :
π − x∗ = δ(π − δx∗ )
π − δπ = x∗ − δ 2 x∗
(1 − δ)π = (1 − δ 2 )x∗
(1 − δ)π
x∗ =
1 − δ2
(1 − δ)π
x∗ =
(1 + δ)(1 − δ)
π
x∗ =
= y∗
1+δ
18
So the first mover (let’s say A) receives
uA =
π
1+δ
and the second mover (B) receives
uB = π −
δπ
π
=
1+δ
1+δ
and the game ends at t = 0.
Note that for δ = 1 the SPNE involves an even split (x = π/2).
3.3
Bargaining and Property Law
Let’s return to the basic model of property law in Subsection 2.1. Consider the slightly revised
payoff functions for the rancher A and the famer B, where wA , wB are the payment to each party
and x ∈ {0, 1} is whether or not the fence is built.
uA = ωA − ax + wA
uB = ωB + bx + wB .
Remember from above that if a > b or under Regime 2 bargaining wasn’t necessary. So assume
Regime 1 and that b > a. The size of the pie is b−a, so the feasibility constraint is wA +wB = b−a.
The disagreement outcomes, where x = 0, are dA = ωA and dB = ωB .
3.3.1
Nash Bargaining
The Nash bargaining problem is
max (uA − dA )(uB − dB )
uA ,uB
The participation constraint for A is
ωA + wA ≥ ωA
The participation constraint for B is
ωB + wB ≥ ωB − b
19
We can rewrite the Nash bargaining problem in terms of wA and wB :
max (ωA + wA − ωA )(ωB + wB − ωB
wA ,wB
simplify:
max (wA )(wB )
wA ,wB
subject to
wA + wB = b − a
Substitute:
(b − a − wB )(wB )
Expand:
2
bwB − awB − wB
Differentiate:
b − a − 2wB = 0
b−a
wB =
2
Which means
wA =
b−a
2
That is, an even split of the surplus.
3.3.2
Alternating Offers
Now take the alternating-offers approach. B proposes a contract (x, w). The “pie” to be
divided is π = b − a. B proposes
w∗ = a +
δ(b − a)
1+δ
which A accepts, from the argument in 3.1, and builds the fence. The payoffs are
δ(b − a)
)−a
1+δ
δ(b − a)
= ωA +
1+δ
uA = ωA + (a +
20
and
δ(b − a)
)+b
1+δ
b−a
= ωB +
1+δ
uB = ωB − (a +
If there are zero bargaining costs (δ = 1), there is an even split of the surplus from building the
fence: w = a + b−a
. For positive bargaining costs (δ < 1), B gets a first-mover advantage and
2
gets more than half the surplus. The social optimum is still chosen in equilibrium (the fence is
built only when b > a); adding this type of bargaining costs only affects the division of the surplus
through the first-mover advantage.
21
3.4
Bargaining and Contract Law
Now let’s look at our model of contracting from Subsection 2.2 in light of bargaining. As stated,
the transaction is mutually beneficial for (x, w) such that w ≥ ax and b(x) ≥ w, which can be
written as ax ≤ w ≤ b(x).
3.4.1
Alternating Offers
The interesting implication of adding a bargaining step to contracting is that it affects the division
of the surplus. The party to make the first offer gets a better deal. In particular, the surplus from
contracting in this model is is π = b(x∗ ) − ax∗ . Applying the alternating-offers paradigm, if buyer
B moves first he proposes
δ(b(x∗ ) − ax∗ )
w∗ = ax∗ +
1+δ
and the seller accepts. With perfect enforcement of contracts, seller chooses quality level x = x∗
and the payoffs are
δ(b(x∗ ) − ax∗ )
uA =
1+δ
b(x∗ ) − ax∗
1+δ
The buyer gets a higher payoff for δ < 1. Conversely, if the seller A makes the offer, he gets a
larger share of the surplus. What are the new participation constraints?
uB =
3.4.2
Nash Bargaining
As a final example on Nash bargaining, let’s assume that the buyer’s utility function is uB =
b(x − w) and that the disagreement outcomes are dA > 0, dB > 0 rather than zero.
The Nash bargaining solution solves:
max(w − ax − dA )(b(x − w) − dB )
w
Multiplying out this expression gives
wb(x − w) − axb(x − w) − dA b(x − w) − wdB + axdB − dA dB
22
And taking first-order conditions for w:
b(x − w) − wb0 (x − w) + axb0 (x − w) + da b0 (x − w) − dB = 0
b(x − w) − dB − (w − ax − dA )b0 (x − w) = 0
b(x − w) − dB = (w − ax − dA )b0 (x − w)
which characterizes the Nash bargaining solution for w.
23
4
Property Law
4.1
Coase Theorem
O’Flaherty 2005, Chapter 7 Section V:
• The Coase theorem means that, in arguing for a government intervention, you have to show
not only that what someone does affects someone else, but also that the parties cannot
negotiate themselves out of the problem.”
• The Coase theorem also suggests another type of policy, aside from taxes, subsidies, or tolls,
to alleviate the problems caused by externalities—helping the parties negotiate.
• Lawyers sometimes refer to property as a “bundle of sticks.” By that they mean that ownership is really a collection of rights. What we call “owning a tire iron” consists of the right
to keep other people from using it, the right to carry it in your trunk, the right to change
your own tires with it, ... and so on. These are the sticks that are in the bundle. “Owning a
tire iron,” in most of the United States today, does not include the right to use it to remove
other people’s tires unless they say you can, ... the right to throw it at imaginary rabbits on
a crowded sidewalk, ... and so on. These are some of the sticks not in the bundle labeled
“owning a tire iron.”
• The Coase theorem says that, so far as efficiency is concerned, if people can negotiate,
it doesn’t matter which sticks are in which bundle. But every stick has to be assigned
unambiguously to one and only one bundle, and it’s the government’s job to assign them.
• In more technologically dynamic arenas like the Internet, organ transplants, and the deep
seabed, sticks that are in no bundle are constantly turning up. Efficient negotiation can
take place only after those sticks have found a home in some bundle.
4.2
Poorly Defined Property Rights
Recall the basical model of property rights with a landowner A and his neighbor, B. A chooses
a binary action, x ∈ {0, 1}. A and B begin with endowments ωA and ωB . A monetary transfer
between the parties is represented by w. The cost of x to A is parametrized by a and the benefit
of x to B is parametrized by b. The payoff functions for A and B are
uA = ωA + w − ax
uB = ωB − w + bx.
24
Previously we looked at the cases of Regime 1 (no fence building requirement) and Regime 2
(fences required but exemptions allowed). We showed that the outcome didn’t matter when
negotiation was allowed.
Now consider the case where the property rights are not well-defined. There is a propability
p that a court will follow Regime 1, and a probability 1 − p that a court will follow Regime 2. In
Regime 2, the Farmer doesn’t want an agreement; in Regime 1, he does.
4.3
Transaction Costs
Recall the basical model of property rights with a landowner A and his neighbor, B. A chooses
a binary action, x ∈ {0, 1}. A and B begin with endowments ωA and ωB . A monetary transfer
between the parties is represented by w. The cost of x to A is parametrized by a and the benefit
of x to B is parametrized by b.
Now extend the model to allow for a cost γ ≥ 0 of writing and enforcing an agreement about
the fence. When there is an agreement, both parties have to pay γ; without an agreement they
don’t pay the cose. The payoff functions for A and B without bargaining are
uA = ωA + w − ax
uB = ωB − w + bx
and with bargaining are
uA = ωA + w − ax − γ
uB = ωB − w + bx − γ.
Recall that the case where we have bargaining is Regime 1, where b > a. Now we require that
b − 2γ > a in order for a bargain to occur. For b < a, no bargain occurs and that’s efficient. For
b > a + 2γ, the transaction occurs and its an efficient outcome. For b ∈ (a, a + 2γ), no bargain
occurs due to transaction costs, but the outcome is inefficient. In that range of b, moving to
Regime 2 improves efficiency.
25
5
Tort Law
5.1
Bilateral Precaution Model
As discussed in the slides, the various doctrines of tort liability put more or less incentive pressure
on the injurer and victim. Here we will give a formal model of tort liability in which both the
plaintiff (victim) and the defendant (injurer) can set precaution levels.
The parties are the defendant A and the plaintiff B. The defendant chooses precaution x and
the plaintiff chooses precaution y. The probability of the accident is given by
p(x, y) = p(x) + q(y)
The harm from the accident is given by H. Here we use cost functions rather than utility functions:
cA = x

y
cB =
y − H
no accident
accident
To minimize the social cost of the accident, one would minimize the expected social cost
SC = E(cA + cB ) = x + y + p(x, y)H
= x + y + [p(x) + q(y)]H
The socially optimal level x∗ of the injurer’s precaution is obtained by the derivative for x:
1 = −p0 (x∗ )H
and similarly for y ∗ :
1 = −q 0 (y ∗ )H
Suppose that the injurer has to pay damages D in the event of an accident. The expected
cost functions are
E(cA |x) = x + [p(x) + q(y)]D
and
E(cB |y) = y + [p(x) + q(y)](H − D)
26
The case of D = 0 is the case of no liability. The injurer solves
min x
x≥0
and chooses x = 0. The victim solves
min y + q(y)H
y≥0
chooses the social optimum y = y ∗ such that 1 = −p0 (y ∗ )H. The no liability regime is good for
victim’s incentives but not for injurer’s incentives.
The case of D = H is the case of strict liability. The injurer solves
min x + p(x)H
x≥0
and chooses the social optimum x = x∗ . The victim solves
min y
y≥0
and chooses y = 0. Strict liability is good for the injurer’s incentive but bad for the victim’s
incentive.
Now consider a simple negligence rule. The injurer is liable if x < x̃, where x̃ is the negligence
standard – the minimum precaution level specified by tort law below which the injurer will be
held negligent in the event the harm occurs. Ideally, x̃ = x∗ , which we’ll assume for now. The
injurer’s expected cost is now
E(cA |x) =

x + [p(x) + q(y)]H
if x < x∗
x
if x ≥ x∗
27
The two possible outcomes are x = 0 or x = x∗ . If x = 0, the victim is insured and chooses
y = 0. The expected utilities are
E(cA |x = 0) = [p(0) + q(0)]H
and
E(cA |x = x∗ ) = x∗
If [p(0) + q(y ∗ )]H > x∗ , the injurer chooses the social optimum. Given x = x∗ , the victim faces
the full costs of the injury:
E(cB ) = y + [p(x) + q(y)]H
and will choose y = y ∗ .
Next consider contributory negligence. The injurer will be treated as negligent and held
responsible for the harm if he chooses x < x̃, but not if y < ỹ, where ỹ is the precaution
standard for the plaintiff. If ỹ ≤ y ∗ , nothing changes relative to the negligence case because given
the assumptions on the production function (that is, perfect substitutability in precaution), the
plaintiff chooses y ∗ anyway.
We can model comparative negligence in the following intuitive way. The injurer is liable for
the full cost of the harm H if x < x̃ and y ≥ ỹ, but proportionally liable if x < x̃ and y < ỹ. An
intuitive assignment rule would have the injurer pay
D(x, y) =
x̃ − x
x̃ − x + ỹ − y
28
which is basically the ratio of negligence levels between the parties. If we have x̃ = x∗ , ỹ = y ∗ ,
the victim’s cost function given x = x∗ is
E(cA ) = y + (p(x∗ ) + q(y))H
so he chooses y = y ∗ . Similarly, the injurer’s expected cost given y = y ∗ is
E(cA |x) =

x + [p(x) + q(y)]H
if x < x∗
x
if x ≥ x∗
and he will choose x = x∗ . So comparative negligence also has the same outcome as negligence
under these assumptions.
This is the key point: given perfect compensation and a legal standard equal to the efficient
levl of care, every form of negligence rule gives the injurer and victim incentives for efficient
precaution.
5.2
Separable Precaution
Assume p(x) = α1 (1 − x)α and q(y) = β1 (1 − y)β where α > 1,β > 1. Find the optimal negligence
standards x̃ and ỹ. Find the chosen precaution levels, probability of accident, and the expected
utilities under no liability, strict liability, negligence, contributory negligence, and comparative
negligence.
Solution.
SC = E(cA + cB ) = x + y + p(x, y)H
1 1
1 1
= x + y + [ ( − x)α + ( − y)β ]H
α 2
β 2
The socially optimal precautions are obtained by
1
0 = 1 − ( − x)α−1 H
2
1
1 α−1
1
( )
=
−x
H
2
1
1
x∗ =
− H 1−α
2
29
and
1
0 = 1 − ( − y)β−1 H
2
1
1 β−1
1
( )
=
−y
H
2
1
1
y∗ =
− H 1−β
2
Note that both x∗ and y ∗ are increasing with H because α > 1, β > 1.
Under no liability, we have
xN L = 0
yN L = y ∗
Under strict liability:
xN L = x∗
yN L = 0
Under negligence, two possible injurer choices are x = 0 or x = x∗ . If x = 0, the victim is insured
and chooses y = 0.
The expected utilities are
E(cA |x = 0) = [p(0) + q(0)]H
1 1
1 1
= [ ( )α + ( )β ]H
α 2
β 2
H
H
=
+ β
α
α2
β2
and
1
1
− H 1−α
2
The condition on the harm level for the injurer to choose x∗ is
E(cA |x = x∗ ) =
1
H
H
1
+ β > − H 1−α
α
α2
β2
2
which we will assume holds. the injurer chooses the social optimum. Given x = x∗ ,
30
5.3
Substitutable Precaution
The probability of harm H is β1 (1 − x − y)β and the costs of precaution are 1 and α, respectively.
x + αy +
1
(1 − x − y)β H
β
so we have
0 = 1 − (1 − x − y)β−1 H
and
0 = α − (1 − x − y)β−1 H
If α < 1, victim has cheaper precaution and victim should take precaution; if α > 1, victim has
more expensive precaution and injurer should take precaution. Say y = 0, then
0 = 1 − (1 − x)β−1 H
(1 − x)β−1 H = 1
1
1 − x = H − β−1
1
x∗ = 1 − H − β−1
Similarly if x = 0 then
1
y ∗ = α − H − β−1
With costless bargaining between the parties, they will figure out α and write a contract specifying
who should take precaution. But otherwise, both parties would want to be the one taking no
precaution.
No liability results in x = 0, y = y ∗ . This is optimal if α < 1.
Strict liability results in x = x∗ , y ∗ = 0. This is optimal if α > 1.
Negligence with x̃ = x∗ results in x = x∗ . Then expected costs for y is
αy +
1
1
1
(1 − x − y)β H = αy + (1 − (1 − H − β−1 ) − y)β H
β
β
1
1
= αy + (H − β−1 − y)β H
β
31
with FOC
1
0 = α − (H − β−1 − y)β−1 H
1
α
(H − β−1 − y)β−1 =
H
1
α 1
− β−1
H
− y = ( ) β−1
H
1
α 1
y = H − β−1 − ( ) β−1
H
which could be greater than zero, in which case the precaution levels would be higher than the
social optimum.
Contributory negligence with ỹ = y ∗ results, symmetrically, with
1
x = H − β−1 − (
1
1 β−1
)
H
Contributory negligence is preferred to simple negligence when α < 1.
5.4
Court Errors
The parties are the defendant A and the plaintiff B. The defendant chooses precaution x and
the plaintiff chooses precaution y. The probability of the accident is given by
p(x, y) = p(x) + q(y)
The harm from the accident is given by H. Here we use cost functions rather than utility functions:
cA = x

y
cB =
y − H
no accident
accident
In the event of a lawsuit, the court makes the correct decision with probability α < 1.
With a negligence standard, the court finds the negligence standard to be x∗ + , where is
uniformly distributed between [−β, β].
5.5
Vicarious liability 1
The players are an employee E, an employer P , and a victim V . The employee chooses precaution
x, the employer chooses precaution y. The employee has assets A. The probability of the accident
32
is
p(x, y)
The administrative costs of filing suit against a party is given by C. Let the number of people
sued be given by n. The social costs are
SC = x + y + p(x, y)H + nC
The social optimum is
−1 = px H
−1 = py H
n = 0
Under no liability, or if not sued, the employee and employer choose x = 0 and y = 0. Social
costs are p(0, 0)H. Under strict liability for the employee, we have
Example 1. Let H = 10, and
1
1
p(x, y) = (1 − x)2 + (1 − αy)2
2
2
The social costs are
1
α
SC = x + y + [ (1 − x)2 + (1 − y)2 ]10 + nC
2
2
Optimality means
1 = 10(1 − x)
1
= 1−x
10
9
x∗ =
10
1 = 10α(1 − y)
1
= 1−y
10α
y∗ = 1 −
and n∗ = 0.
33
1
10α
Without liability, we have x = 0, y = 0, n = 0.
With strict liability and lawsuits against both parties, we have x∗ and y ∗ .
5.6
Vicarious liability 2
The players are an employee, employer, and victim. The employee chooses precaution x and has
assets A. The principal can monitor the employer’s precaution and offer a wage w + bx, where
b ≥ 0 is the payment for precaution. The probability of the accident is p(x). The social costs are
SC = x + p(x)H
The social optimum is
−1 = px H
Under no liability, the employee chooses x = 0 and the employer chooses b = 0. Social costs are
p(0, 0)H. Under strict liability for the employee, we have
1
Example 1. Let H = 10 and p(x) = 1+x
. The social optimum is
0 = 1 − (1 + x)−2 4
1
= (1 + x)−2
4
2 = 1+x
x = 1
without liability we have x = 0.
With strict liability we have employee costs
(1 − b)x + A(1 + x)−1
which is solved as
1 − b = A(1 + x)−2
A 1
(
)2 = 1 + x
1−b
A 1
x = (
)2 − 1
1−b
Without vicarious liability, the employer will set b = 0 and we get x =
34
√
A − 1.
We can reach the social optimum by setting b such that
A 1
)2 − 1
1−b
A 1
(
)2
1−b
A
1−b
A
4
A
1−
4
1 = (
2 =
4 =
1−b =
b =
The lower the employee’s assets, the higher the wage needed to induce compliance. If A = 4, no
need for vicarious liability.
35
6
Contract Law
6.1
Efficient breach and efficient reliance example
These are notes based on Cooter and Ulen, Chapter 9 Appendix.
• Xavier promises to perform a service for Yvonne, for example expanding her store. He
spends x on performing, and the probability of performance is p(x).
• Yvonne relies on the contract, for example by expanding inventory. The reliance expenditures
are given by y. Yvonne’s payoff goes up with reliance and is complementary with whether
performance occurs:
• Efficiency requires choosing x and y to maximize Yvonne’s expected profits minus Xavier’s
expenditures. Formally:
max p(x)F (y) + (1 − p(x))G(y) − x − y
x,y
FOC for x is
1 = px [F (y) − G(y)]
FOC for y is
1 = pFy + (1 − p)Gy
These equations determine the optimal x∗ and y ∗
36
• Expectations damages DE puts Yvonne in the same position as if Xavier performed, that is
profits with performance minus profits without performance:
DE = (F (y) − y) − (G(y) − y)
= F (y) − G(y)
• Reliance damages DR puts Yvonne in the same position as if she had not signed a contract.
Assume that without contract her reliance expenditures would have been y0 and her profits
would have been G(y0 ) − y0 . So her reliance damages for reliance level y would be
DR = (G(y0 ) − y0 ) − (G(y) − y)
and we know that DR < DE
• We could also compute opportunity-cost damages if we specified the terms of a next-best
contract.
• Given damages D, Xavier solves
min x + (1 − p(x))D
x
which gives
1 = px D
Under expectations damages, we have
1 = px [F (y) − G(y)]
Under reliance damages we have
1 = px [G(y0 ) − y0 ) − (G(y) − y)]
Expectations damages is optimal: xE = x∗ ; because DR < DE , reliance damages give
inefficiently low effort incentives: xR < x∗ .
• Given damages D, Yvonne solves
max pF (y) + (1 − p)[G(y) + D(y)] − y
y
37
• Under perfect expectations damages (which uses y ∗ ), Yvonne maximizes
pF (y) + (1 − p)[G(y) + F (y ∗ ) − G(y ∗ )] − y
which results in privately optimal reliance
1 = pFy + (1 − p)Gy
which is efficent.
• Under imperfect expectations damages (which uses y), Yvonne maximizes
pF (y) + (1 − p)[G(y) + F (y) − G(y)] − y = pF (y) + (1 − p)[F (y)] − y
= F (y) − y
which results in reliance
1 = Fy
Which is suboptimally high
• Under reliance damages, Yvonne maximizes
pF (y) + (1 − p)[G(y) + (G(y0 ) − y0 ) − (G(y) − y)] − y = pF (y) + (1 − p)[G(y0 ) − y0 + y)] − y
= pF (y) + (1 − p)[G(y0 ) − y0 ] − py
which results in reliance
1
= Fy
p
which is even higher than imperfect damages.
• Finally, under no damages (or restitution), we would have Yvonne maximizing
pF (y) + (1 − p)[G(y)] − y
which also results in privately optimal reliance
1 = pFy + (1 − p)Gy
6.2
Another model
In contract theory, A can be thought of as the seller, and B the buyer.
38
With externalities, introduce C as a third party.
Consider a seller A and a buyer B. A promises to sell to B a good with quality q at price p,
but B will observe the quality only after paying p. The cost to A of q is c(q), which is increasing
and convex. So the profit from the transaction for A is
πA (p, q) = p − c(q).
The buyer’s utility is given by
uB (p, q) = u(q) − p
where u(·) is increasing and concave. We normalize the outside options for each agent to zero.
In consequence, the transaction is mutually beneficial, and therefore both parties would agree to
it, for any (p, q) such that p ≥ c(q) and u(q) ≥ p, which can be written more succinctly as
c(q) ≤ p ≤ u(q). These are the participation constraints for buyer and seller.
At the social optimum, the marginal benefit of quality equals the marginal cost. The optimal
quality level q ∗ satisfies
u0 (q ∗ ) = c0 (q ∗ )
which, along with any p where c(q∗) ≤ p ≤ u(q ∗ ), specifies the possible social optima.
39
7
Criminal Law
7.1
Model of options for reducing crime
In the intro model of crime, the government could affect crime rates only by increasing the probability of detection. Now we will consider two other policy options: 1) increasing the punishment
for convicted criminals, and 2) subsidizing non-crime activity to increase the opportunity cost of
crime.
The criminal chooses what proportion of time to spend on crime, x ∈ [0, 1]. So the amount of
time spent on non-crime activities is 1 − x. The benefit from crime is given by the function b(x),
which increases concavely with x. The benefit from non-crime is given by (a + s)(1 − x), where
a is a positive real number and s is the government subsidy to non-crime. The punishment for a
crime level x is given by f (x), which is assumed to be increasing in x. Finally, the probability of
detection and punishment is given by p(x), which is also increasing in the amount of time spent
on crime. The expected sanction for crime level x is p(x)f (x).
The criminal’s expected payoff is
E(u(x)) = b(x) + (a + s)(1 − x) − p(x)f (x).
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The derivative for x gives the optimality condition
bx = a + s + px f (x) + p(x)fx
which equates the marginal benefit of crime with the marginal cost.
• Because b(x) is strictly concave, b0 (x) is decreasing, so increasing s will decrease x.
• Similarly, increasing the probability of detection p(x) will decrease x
• Similarly, increasing the sanction f (x) will decrease x.
To see this more clearly, assume that y is the expenditure on police and z is the expenditure on
prisons, where now we have p(x, y) and f (x, z). The criminal’s new expected utility is
E(u(x)) = b(x) + (a + s)(1 − x) − p(x, y)f (x, z).
With first-order condition
bx = a + s + px (x, y)f (x, z) + p(x, y)fx (x, z)
1
Example. Let b(x) = 31 x 3 , u(x) =
have
a+s
(1
3
1
2
1
2
b x = x− 3
2
ux = −(a + s)(1 − x)− 3
1
1
p x = x− 3 y 2
2 1 1
f x = x− 3 z 2
3
The criminal’s optimal crime is defined by
2
1
1
2 1
2
3 2 1 2 1 1
x− 3 = (a + s)x− 3 + (x− 3 y 2 )(x 3 z 2 ) + ( x 3 y 2 )( x− 3 z 2 )
2
3
1
1 1
1
1 1
− 32
− 32
x
= (a + s)x + x 3 y 2 z 2 + x 3 y 2 z 2
2
1
1
1
1
1
2
1
− x) 3 ,p(x, y) = 32 x 3 y 2 , and f (x, z) = x 3 z 2 . We
(1 − a − s)x− 3 = 2x 3 y 2 z 2
2y 2 z 2
x
=
(1 − a − s)
(1 − a − s)
x =
1 1
2y 2 z 2
−1
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8
Constitutional Law
This section discusses Maskin and Tirole (AER, 2004).
Under RD, the official chooses a with probability pπ +(1−p)(1−π). The posterior probability
that the politician is congruent given the choice of a is
pπ
pπ + (1 − p)(1 − π)
This is also the expected utility from retaining the politician. The probability the official chooses
b is p(1 − π) + (1 − p)π. The official is removed in this case and another official is elected, who
is congruent with probability π and gives an expected utility of π. Therefore the expected
welfare in the second period is
[pπ + (1 − p)(1 − π)][
pπ
] + [p(1 − π) + (1 − p)π][π]
pπ + (1 − p)(1 − π)
which simplifies to
pπ + [p(1 − π) + (1 − p)π]π
which we add to the first period payoff to get the expected utility for representative democracy:
π + pπ + [p(1 − π) + (1 − p)π]π
which is strictly better than judicial power (greater than 2π). It is better than direct democracy when
π + pπ + [p(1 − π) + (1 − p)π]π > 2p
π + pπ + pπ − pπ 2 + π 2 − pπ 2 > 2p
π + π 2 > p(2 − 2π + 2π 2 )
π(1 + π)
p <
2(1 − π + π 2 )
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